高数问题求解‼lim(tanx-sinx)／(sinx)^3 .x趋向0

高数问题求解‼lim(tanx-sinx)／(sinx)^3 .x趋向0
高数问题求解‼lim(tanx-sinx)／(sinx)^3 .x趋向0

高数问题求解‼lim(tanx-sinx)／(sinx)^3 .x趋向0
lim(x→0) (tanx - sinx)/(sinx)^3
= lim(x→0) (tanx - sinx)/x^3,分母等价无穷小
= lim(x→0) sinx(1/cosx - 1)/x^3
= lim(x→0) (sinx)/x * (1 - cosx)/(x^2 * cosx)
= lim(x→0) 2[sin(x/2)]^2/x^2
= lim(x→0) [sin(x/2)]^2/(x/2)^2 * 1/2
= 1/2

原式=lim(x→0)(1/cosx-1)/(sinx)^2=lim(x→0)(1-cosx)/(sinx)^2*1/cosx=lim(x→0)2(sin(x/2))^2/(sinx)^2=lim(x→0)2(x/2)^2/x^2=1/2

tanx-sinx等价于(1/2)x^3。记住吧，很有用的。tanx,x,sinx.任意两个相减都是等价于x^3，tanx-x~(1/3)x^3。x-sinx~(1/6)x^3。当然都是x趋向0.可以用迈克劳林展开的方法求证